Is a sport relevant? No. A competition of any kind? Yesish. Did 10 win something? No, 10 lost something. Did 20 lose something? No, 20 won something. If so: money? possessions? a ranking? freedom? life? This one, for both.

So 10 is a number that represents one person (e.g. the tenth person in line, or subject #10)? And the same for 20?

Are these numbers that represent the people assigned/chosen by others? by themselves? Are the number assigners researchers? medical professionals? teachers? Are both of the other two relevant people (I'll call them Pat and Morgan here for clarification purposes, if that's okay) the number assigners? Is either (one of) the number assigner(s)?

Were 10 and 20 assigned these numbers by virtue of when they got somewhere or started or signed up for something? by virtue of where they were but not necessarily when they got there (e.g. seats 10 and 20)? at random? relevant?

So 10 is a number that represents one person Yes. (e.g. the tenth person in line, or subject #10 But not those. The numbers are NOT a place in a sequence.)?And the same for 20? Yes.

Are these numbers that represent the people assigned/chosen by others? These ones. by themselves? Are the number assigners researchers? medical professionals? teachers? None of these. Are both of the other two relevant people (I'll call them Pat and Morgan here for clarification purposes, if that's okay) the number assigners? No. Is either (one of) the number assigner(s)? No, neither of them are.

Were 10 and 20 assigned these numbers by virtue of when they got somewhere or started or signed up for something? by virtue of where they were but not necessarily when they got there (e.g. seats 10 and 20)? Yope. at random? Yope. relevant? Only in the sense that figuring out how these numbers were assigned will help you determine who the people were, and that the numbers provide a (somewhat lateral) connection between them.

Were the numbers 10 and 20 assigned to their respective people at essentially the same time? Were numbers assigned to more than just the two numbered in the puzzle statement (10 and 20)? Did Pat and Morgan have numbers as well? Did anyone other than these four have numbers assigned to them?

Were the numbers assigned all even numbers? Was the number 1 ever assigned? 25? 26? 27? Anything higher than 99?

Were the numbers 10 and 20 assigned to their respective people at essentially the same time? Essentially. (I think 20 may have been assigned slightly before 10, but it's not particularly important.) Were numbers assigned to more than just the two numbered in the puzzle statement (10 and 20)? Yes. Did Pat and Morgan have numbers as well? No. Did anyone other than these four have numbers assigned to them? Yes.

Were the numbers assigned all even numbers? No. Was the number 1 ever assigned? Yes. 25? No. 26? No. 27? No. Anything higher than 99? Yes.

Were the numbers chosen before people started being assigned them? Were the numbers chosen at random? or was there a method to picking them?

Were the chosen numbers assigned randomly to the people who were given numbers? Was there a method for assigning them (e.g. when I was in sports as a child, the larger numbers were on the bigger jerseys)? Could the person who was assigned #20 just as easily have been assigned #10? Vice versa? Would it help us to know how many people in total were assigned numbers? If so, [LTPF list of integers greater than 2]?

Was there only one person or entity assigning numbers? More than one? Did the people who were assigned numbers actually assign their own numbers? The numbers of others among those people assigned numbers?

Were the numbers chosen before people started being assigned them? Yes. Were the numbers chosen at random? or was there a method to picking them? This one.

Were the chosen numbers assigned randomly to the people who were given numbers? Probably not, but it's not particularly relevant. Was there a method for assigning them (e.g. when I was in sports as a child, the larger numbers were on the bigger jerseys)? Irrel. Could the person who was assigned #20 just as easily have been assigned #10? Yes. Vice versa? Yes. Would it help us to know how many people in total were assigned numbers? If so, [LTPF list of integers greater than 2]? Seven numbers are currently assigned. More have been assigned in the past, but their numbers are no longer current.

Was there only one person or entity assigning numbers? One entity. More than one? Did the people who were assigned numbers actually assign their own numbers? No. The numbers of others among those people assigned numbers? No.

***********************RECAP*********************

One person, assigned number 10, lost a competition of some sort which caused him to lose his life. Another person, assigned number 20, won a similar competition, with the result being his survival. Two other people, Pat and Morgan, were involved in these competitions as well. All four involved are H, A, M and dead.

Various people besides 10 and 20 have been assigned numbers over the years. Currently, there are 7 of them, although there have been more in the past. The number 1 is currently assigned, but 15, 25, 30, and 40 have never been assigned at any time. Pat and Morgan have never been assigned numbers.

So were Pat and Morgan the entity that assigned numbers? And to clarify all four are currently dead? Did Pat and Morgan's deaths have anything to do with the competition? 10 died from losing the competition? Did 20's death have anything to do with the competition? Do the 7 people assigned numbers all belong to the same organization? Was their participation voluntary? Or forced on them? Once a number is assigned to someone is it ever again assigned to someone else? Or is each number only used once? Is it important to find out why 15, 25, 30 and 40 have never been assigned? Could they ever be assigned in the future? Or are these 4 numbers never to be used for some reason? Does it have something to do with mathematic formulae?

So were Pat and Morgan the entity that assigned numbers? No. And to clarify all four are currently dead? Yes. Did Pat and Morgan's deaths have anything to do with the competition? Morgan's death did; Pat's did not. 10 died from losing the competition? Yes. Did 20's death have anything to do with the competition? No. Do the 7 people assigned numbers all belong to the same organization? Yes, for SVV of "organization." Was their participation voluntary? This one. Or forced on them? Once a number is assigned to someone is it ever again assigned to someone else? It has been, in the past. Or is each number only used once? Only once at a time, but as mentioned above, the numbers have occasionally been reassigned over the years. Is it important to find out why 15, 25, 30 and 40 have never been assigned? It would be useful to find out EXACTLY which seven numbers are currently assigned. Doing that would give you a BIG hint to what the numbers mean.Could they ever be assigned in the future? Theoretically, but it's quite unlikely. Or are these 4 numbers never to be used for some reason? No; the entity assigning them simply chose particular numbers and not others. Does it have something to do with mathematic formulae? Noish.

Is this competition periodic? Is it held every year? Biennially? Quadrennially? Or some other time period is used (eg. months, or days)?

For how long has this competition been going on? Less than 5 years? Between 5-10 years? More than 10? More than 25? Is it an event confined to one city? one state? one country? one continent? one international organisation (like the Commonwealth, or the Afro-Asian Games)

Or is it international?

Is it relevant that only 7 numbers are assigned?

Are the numbers assigned COMPLETELY at random?

You said 'noish' to math formulae - Is this competition related to mathematics? Science? IQ testing? Calculation competitions? Mathletes?

Is this competition physical? Mental? A bit of both?

Can negative numbers be assigned? Can 0 be assigned? Can non-integers be assigned?

Can I ask which numbers were assigned (using 'LTPF list of integers') or would that be too much?

Is this competition periodic? No. Is it held every year? Biennially? Quadrennially? Or some other time period is used (eg. months, or days)? So, no.

For how long has this competition been going on? Less than 5 years? Between 5-10 years? More than 10? More than 25? Competitions of this sort have been held throughout history, but any individual one lasts only a few minutes.Is it an event confined to one city? one state? one country? one continent? one international organisation (like the Commonwealth, or the Afro-Asian Games) None of these. It's organized only by the two participants themselves.

Or is it international? So, no.

Is it relevant that only 7 numbers are assigned? Only in the sense that knowing which numbers is likely a BIG clue.

Are the numbers assigned COMPLETELY at random? No, but not particularly relevant.

You said 'noish' to math formulae - Is this competition related to mathematics? Science? IQ testing? Calculation competitions? Mathletes? None of these. Change that "noish" to "no."

Is this competition physical? Yesish. Mental? A bit of both? No to rest.

Can negative numbers be assigned? No. Can 0 be assigned? No. Can non-integers be assigned? No.

Can I ask which numbers were assigned (using 'LTPF list of integers') or would that be too much? Perhaps, but I'll tell you anyway: 1, 2, 5, 10, 20, 50, and 100 are currently assigned. Other numbers have been assigned in the past.

Is losing your life a common result of losing the competition? YES!

If 10 had decided not to participate in the competition - would he have lost his life? No. If 20 had decided not to paricipate in the competition - would he have survived? Well, he wouldn't still be alive now, because he lived more than a century ago. He died of natural causes, and would have anyway, of course.

Is the "competition" a duel? YES! With Pat and Morgan being seconds? But no. OTRT, though.

Is it at all possible to lose the competition/duel and survive? Depends how you define "lose," but I would say no.

...are the numbers coin values? No, but you're close enough that I'll go ahead and tell you. They're the values of currently circulated American dollar bills (if you're not an American, that may not be obvious).

Were the individuals assigned minted coins? the values? The individuals are depicted on American paper currency.Were the individuals to be involved anyway with the design of future coins? No.Was a poor design punished by death? No.

Would the competition be likely to take place in the present? It could still happen, but it's far less common than it used to be. In western civilisation? It could. Would it still commonly result in death? Yes.

Which of these are relevant? Andrew Jackson? Alexander Hamilton? Aaron Burr? All of them are.

Are Pat and Morgan the opponents of 10 and 20? Yes, they are.

Which means it's time for a

***********************SPOILER*******************

Alexander Hamilton (the first U.S. Secretary of the Treasury, depicted on the U.S. $10 bill) was killed in an 1804 duel with sitting Vice President Aaron Burr (thus, "10 lost"). Two years later, future president Andrew Jackson (depicted on the $20 bill) killed Charles Dickinson in a duel after the latter slandered his wife, Rachel (thus, "20 won").

Thanks for playing! I'll try to have a new puzzle up as soon as I can.

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